Definition for continuous flows

Comments

There is some variation in the definition across various publications;

Historically, the first case considered was that of a continuous flow on the plane, induced by an ordinary differential equation. In this case, a homoclinic connection is a single trajectory that converges to the fixed point p{\displaystyle p} both forwards and backwards in time. A pendulum in the absence of friction is an example of a mechanical system that does have a homoclinic connection. When the pendulum is released from the top position (the point of highest potential energy), with infinitesimally small velocity, the pendulum will return to the same position. Upon return, it will have exactly the same velocity. The time it will take to return will increase to ∞{\displaystyle \infty } as the initial velocity goes to zero. One of the demonstrations in the pendulum article exhibits this behavior.

Significance

When a dynamical system is perturbed, a homoclinic connection splits. It becomes a disconnected invariant set. Near it, there will be a chaotic set called Smale's horseshoe. Thus, the existence of a homoclinic connection can potentially lead to chaos. For example, when a pendulum is placed in a box, and the box is subjected to small horizontal oscillations, the pendulum may exhibit chaotic behavior.